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[[File:Dual cone illustration.svg|right|thumb|A set <math>''C</math>'' and its dual cone <math>''C^*</math>''.]]
[[File:Polar cone illustration1.svg|right|thumb|A set <math>''C</math>'' and its polar cone ''C<mathsup>C^o</mathsup>''. The dual cone and the polar cone are symmetric to each other with respect to the origin.]]
'''Dual cone''' and '''polar cone''' are closely related concepts in [[convex analysis]], a branch of [[mathematics]].
==Dual cone==
The '''dual cone''' <math>''C^* </math>'' of a [[subset]] <math>''C</math>'' in a [[linear space]] <math>''X</math>'', e.g. [[Euclidean space]] '''R'''<mathsup>\mathbb R^''n''</mathsup>, with [[topological]] [[dual space]] <math>''X^*</math>'' is the set
:<math>C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},</math>
where <math>\langle ⟨''y'', ''x \rangle</math>''⟩ is the duality pairing between ''X'' and ''X*'', i.e. ⟨''y'', ''x''⟩ = ''y''(''x'').
between <math>X</math> and <math>X^*</math>, i.e. <math>\langle y, x \rangle = y(x) </math>.
<math>''C^* </math>'' is always a [[convex cone]], even if <math>''C </math>'' is neither [[convex set|convex]] nor a [[linear cone|cone]].
Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as '''R'''<mathsup>\mathbb{R}^''n''</mathsup> equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.
:<math>C^*_{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}.</math>
whenUsing <math>this latter definition for ''C *'', </math>we have that when ''C'' is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization | first1=Stephen P. |last1=Boyd |first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3 | url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref> ▼Using this latter definition for <math>C^*</math>, we have that
* A non-zero vector ''y'' is in ''C*'' if and only if both of the following conditions hold:
▲when <math>C </math> is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref>
* A non-zero vector <math>#''y</math> is in <math>C^*</math> if and only if both of the following conditions hold: (i) <math> y </math>'' is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] <math>''C </math>''. (ii) <math> y </math> and <math>C </math> lie on the same side of that supporting hyperplane.
#''y'' and ''C'' lie on the same side of that supporting hyperplane.
*<math>''C^* </math>'' is [[closed set|closed]] and convex.
*''C''<mathsub>1</sub>C_1 \subseteq⊆ C_2''C''<sub>2</mathsub> implies <math>C_2^* \subseteq C_1^*</math>.
*If <math>C </math> has nonempty interior, then <math>C^* </math> is ''pointed'', i.e. <math>C^* </math> contains no line in its entirety.
*If <math>''C'' </math>has isnonempty ainterior, cone and the closure ofthen <math>''C </math>*'' is ''pointed'', theni.e. <math>''C^*'' </math>contains hasno nonemptyline interiorin its entirety.
*<math>If ''C^{**} </math>'' is a cone and the closure of the''C'' smallestis convexpointed, conethen containing <math>''C*'' has nonempty </math>interior.
*''C**'' is the closure of the smallest convex cone containing ''C''.
==Self-dual cones==
A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an [[inner product]] ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to <math>''C </math>''.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref> Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual. This is slightly different than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in '''R'''< mathsup> \mathbb{R}^''n ''</ mathsup> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base in '''R'''<sup>''n''</sup> is equal to its internal dual.▼
The nonnegative [[orthant]] of '''R'''< mathsup> \mathbb{R}^''n ''</ mathsup> and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in '''R'''< mathsup> \mathbb{R}^3</ mathsup> whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in '''R'''< mathsup> \mathbb{R}^3</ mathsup> whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square. ▼A cone <math>C</math> in a vector space <math>X</math> is said to be ''self-dual'' if
<math>X</math> can be equipped with an [[inner product]]
<math>\langle . , . \rangle</math> such that the
internal dual cone relative to this inner product
▲is equal to <math>C</math>.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref> Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual. This is slightly different than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in <math>\mathbb{R}^n</math> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base
in <math>\mathbb{R}^n</math> is equal to its internal dual.
▲The nonnegative [[orthant]] of <math>\mathbb{R}^n</math> and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in <math>\mathbb{R}^3</math> whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in <math>\mathbb{R}^3</math> whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
==Polar cone==
[[File:Polar cone illustration.svg|right|thumb|The polar of the closed convex cone <math>''C</math>'' is the closed convex cone ''C<mathsup>C^o,</mathsup>'', and vice-versa.]]
For a set ''C'' in ''X'', the '''polar cone''' of ''C'' is the set<ref name="Rockafellar">{{cite book|author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis | publisher=Princeton University Press |___location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|pages=121–122}}</ref>
For a set <math>C</math> in <math>X</math>, the '''polar cone''' of <math>C</math> is the set
:<math>C^o = \left \{y\in X^*: \langle y , x \rangle \leq 0 \quad \forall x\in C \right \}.</math><ref name="Rockafellar">{{cite book|author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis|publisher=Princeton University Press|___location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|pages=121–122}}</ref>
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. ''C<mathsup>C^o=-C^*</mathsup>'' = −''C*''.
For a closed convex cone <math>''C</math>'' in <math>''X</math>'', the polar cone is equivalent to the [[polar set]] for <math>''C</math>''.<ref>{{cite book|last=Aliprantis |first=C.D.|last2=Border |first2=K.C. |title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=215}}</ref>
== See also ==
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